Question text 1Mark A manufacturing company wants to estimate the average time it takes to assemble a specific product. A sample of 27 assembly times was collected, with a sample mean of 45 minutes and a standard deviation of 6
minutes. Assume the assembly times are normally distributed. Using a 90% confidence level, the upper limit of the confidence interval for the population mean is Answer 3[input] minutes. State your answer to 2 decimal placesNotes Report question issue Question 13 Notes

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BlackTom AI · Accepted Answer

First, identify what is being asked: an upper limit of the 90% confidence interval for the population mean, given a sample of size 27, sample mean 45 minutes, and sample standard deviation 6 minutes, with the assumption of normal distribution. Since the population standard deviation is unknown and the sample size is relatively small, we should use a t-distribution to form the confidence interval for the mean. The formula for the upper one-sided 90% confidence limit is: upper limit = x̄ + t_{α, df} * (s / √n), where α = 0.10 and df = n − 1 = 26.

Step by step:
- Compute the standard error: s/√n = 6 / √27. The square root of 27 is about 5.196, so the standard error ≈ 6 / 5.196 ≈ 1.1547.
- Look up the t critical value for a one-sided 90% confidence level with 26 degrees of freedom. This corresponds to t_{0.10, 26}. The value is approximately between 1.31 and 1.32 (depending on the exact t table or interpolation). For illustration, use t ≈ 1.315.
- Multiply the t value by the standard error: 1.315 × 1.1547 ≈ 1.519 (approximately).
- Add this to the sample mean to get the upper limit: 45 + 1.519 ≈ 46.519, which rounds to 46.52 when reported to two decimals.

Notes and considerations:
- If one were to use the z-distribution instead of the t-distribution (which is common when σ is known), the one-sided 90% critical value is z_{0.90} ≈ 1.2816. Using that value would yield 45 + 1.2816 × 1.1547 ≈ 45 + 1.481 ≈ 46.48, rounding to 46.48.
- The use of the t critical value is more appropriate here because the population standard deviation is unknown and the sample size is not large. Therefore the t-based upper limit is typically slightly larger than the z-based one.
- If you see an answer like 46.97, that would imply a larger margin of error (approximately 2.0) than what the standard calculation with the given data yields. This could result from using a different t critical value (perhaps misreading the df or using a two-sided interval value by mistake) or a calculator input error. Double-check the t critical value for df = 26 and a one-sided 90% level, as well as the standard error calculation.

In summary, based on the standard approach with t_{0.10, 26} ≈ 1.315 and SE ≈ 1.1547, the upper confidence limit for the population mean is approximately 46.52 minutes (to two decimals).

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